Question Video: Calculating the Scalar Product of Two Vectors Using Unit Vector Notation Physics

Video Transcript

Consider the two vectors 𝐩 equals two 𝐢 plus three 𝐣 and 𝐪 equals six 𝐢 plus four 𝐣. Calculate 𝐩 dot 𝐪.

This representation here of these two vectors tells us that we’re to calculate their scalar or dot product. And we see we’re given the two vectors 𝐩 and 𝐪 in their component form. So, we can start off by recalling that the scalar product of two vectors by their components is equal to the 𝑥-component of the first vector times the 𝑥-component of the second vector. Added to the 𝑦-component of the first vector times the 𝑦-component of the second. In this equation, we’ve called our vectors 𝐀 and 𝐁, but those are just general names for any vectors that lie in the 𝑥𝑦-plane.

In this example, what we want to calculate is 𝐩 dot 𝐪. And to do it, we can follow this prescription for combining the components of these vectors. First, we take the 𝑥-component of our first vector, that’s 𝐩, and the 𝑥-component of that vector is two. And we multiply this by the 𝑥-component of our second vector. That second vector is 𝐪 and that 𝑥-component is six. So, we have two times six. And to that, we add the 𝑦-component of our first vector. That first vector is 𝐩 and that 𝑦-component is three multiplied by the 𝑦-component of our second vector. That second vector is 𝐪 and that 𝑦-component is four.

So what we have then is 𝐩 dot 𝐪 is equal to two times six plus three times four. Two times six is 12 and so is three times four. So, our final answer is 24. And notice that, indeed, this answer is a scalar quantity. It has a magnitude but no direction.